Bayesian neural network with autoencoder for model-based description of $α$-particle preformation factor
$α$ decay is an important probe for studying the structure of heavy and superheavy nuclei, in which the $α$-particle preformation ($P_α$) is a key physical quantity for describing decay half-lives. This work develops a hybrid framework that integrates Bayesian neural networks with autoencoder (BNN-Auto), combined with the cosh potential (CPT), to systematically optimize the constraint and prediction of $P_α$. The model employs variational inference for probabilistic modeling of network weights, naturally providing robust uncertainty quantification for predictions, and utilizes an autoencoder to enhance the robustness of feature representation. Based on experimental data from 535 nuclei, the BNN-Auto method achieves relative improvements in the root mean square deviation ($σ_{\rm{RMS}}$) of $P_α$ prediction by $61.14\%$ on the training set and $54.49\%$ on the validation set. Further analysis reveals that the $P_α$ and half-life extracted by the model exhibit pronounced odd-even staggering and shell effects in isotopic chains with $Z=86-90$ and isotones with $N=124-128$ and $N=150-154$. Moreover, we successfully predict the $α$ decay half-lives of nuclei with $Z=120$ and observe a significant increase in the half-life near $N=184$, which verifies the shell effect of the predicted 'stable island'. This study not only provides a high-precision theoretical description for $α$ decay, but also offers a new machine learning perspective for exploring the structure of superheavy nuclei.